Astrophysics Picsimulation Lyapunov Piclyapunov Cert

The PIC Lyapunov certificate

Particle-in-cell (PIC) simulations are a standard tool in plasma physics: they track many charged particles through their own collective electromagnetic fields, and the Lyapunov exponent measures how quickly nearby trajectories diverge, a practical proxy for chaotic behavior. The Recognition Science framework's library contains a machine-checked certificate, called PICLyapunovCert, that packages three formal facts about a proposed scaling rule for this exponent. The rule is that as the simulation resolution increases, moving from one rung to the next, the Lyapunov exponent shrinks by a factor of the golden ratio φ, roughly 1.618.

Concretely, the certificate fixes a reference value of 1 for the exponent at the coarsest resolution, defines the exponent at higher resolution rung k as φ raised to the power −k, and then proves three properties: the exponent is always positive, each step multiplies the previous value by φ⁻¹ (about 0.618), and the ratio between adjacent rungs is exactly φ⁻¹. These three claims are proved in the framework's machine-checked library of formal theorems, with no unproved assumptions. The certificate itself is a structure that bundles these three proofs together, and the library provides an explicit instance of it, so the whole package is a single object a programmer or physicist could inspect.

The empirical motivation comes from standard PIC literature: Dawson (1983) and Birdsall and Langdon (2004) document that doubling the number of macro-particles per Debye cell reduces numerical heating by a factor of φ². The framework notes this same φ² ratio appears in other contexts, such as Turing pattern thresholds and BCS pairing steps, and treats the PIC agreement as a structural prediction consistent with its broader phi-ladder scheme. That is a claim about a pattern across simulations, not a measurement of a real plasma.

In Recognition Science, this certificate is one small piece of a larger forcing chain that derives constants and structures from a single cost function. But the certificate itself does not prove that real plasmas follow this rule, nor that the golden ratio is the only possible scaling for PIC convergence. It establishes only the internal consistency of a definition: given the proposed formula, the three ratio properties follow. The physical claim, that actual PIC simulations exhibit this exact scaling, remains an empirical check, not a theorem. The certificate is a formal object, a compact summary of what the framework's definitions entail, and its value is precision about what follows from what.

theorem lyapunovAt_pos (k : ℕ) : 0 < lyapunovAt k := by
  unfold lyapunovAt referenceExponent
  have : 0 < phi ^ (-(k : ℤ)) := zpow_pos Constants.phi_pos _
  linarith [this]

theorem lyapunovAt_succ_ratio (k : ℕ) :
    lyapunovAt (k + 1) = lyapunovAt k * phi⁻¹ := by
  unfold lyapunovAt
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  have : phi ^ (-((k : ℤ) + 1)) = phi ^ (-(k : ℤ)) * phi⁻¹ := by
    rw [show (-((k : ℤ) + 1)) = -(k : ℤ) + (-1 : ℤ) by ring]
    rw [zpow_add₀ hphi_ne]; simp
  have hcast : ((k + 1 : ℕ) : ℤ) = (k : ℤ) + 1 := by push_cast; ring
  rw [hcast, this]; ring

theorem lyapunovAt_adjacent_ratio (k : ℕ) :
    lyapunovAt (k + 1) / lyapunovAt k = phi⁻¹ := by
  rw [lyapunovAt_succ_ratio]
  field_simp [(lyapunovAt_pos k).ne']

/-- PIC-simulation Lyapunov certificate. -/
def picLyapunovCert : PICLyapunovCert where
  lyapunov_pos := lyapunovAt_pos
  one_step_ratio := lyapunovAt_succ_ratio
  adjacent_ratio := lyapunovAt_adjacent_ratio

What this page does not claim

The certificate does not prove that real plasma simulations obey the φ-ladder rule. It does not claim that φ is the only possible scaling for PIC convergence. It does not establish that numerical heating in a real PIC code follows the φ² ratio.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Astrophysics/PICSimulationLyapunov.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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